Abstract. We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the q-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the epositivity conjecture of Stanley and Stembridge for incomparability graphs of (3 + 1)-free posets.

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q, p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n − 2)! spheres of dimension 2n − 5. This answers one of the questions raised by Vassiliev [V3] in connection with knot invariants. For this case the S naction on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n − 2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n − 3)-connected graphs we provide a formula for the generating function of the Euler characteristic.1 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 2 There is Z 3 -torsion of rank 8. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 3 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5, 7. 4 There is Z 3 -torsion of rank 35. No Z p -torsion for p = 2, 5, 7. 5 There is Z 3 -torsion of rank 56. No Z p -torsion for p = 2, 5, 7.

Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrécica andŽivaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex M n is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex M n,n is a 3-group of exponent at most 9. When n ≡ 2 mod 3, the bottom nonvanishing homology of M n,n is shown to be Z 3 . Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics. MR Subject Classifications: 05E25, 05E10, 55U10. 12 47 56 37 45 36 67 13 24 15 26 13 24 15 26 13 24 15 26 34 57 Figure 1.1: Piece of matching complex M 7It is conjectured in [BLVZ] that the connectivity bounds of Theorem 1.1 are sharp. The n ≡ 0, 1 mod 3 cases of the conjecture for the matching complex had already been established by Bouc [Bo] who proved the following result. Theorem 1.3 (Bouc[Bo]). (i)H νn (M n ) is finite if and only if n ≥ 7 and n / ∈ {8, 9, 11}. (ii) If n ≡ 1 mod 3 and n ≥ 7 thenH vn (M n ) ∼ = Z 3 . (iii) If n ≡ 0 mod 3 and n ≥ 12 thenH vn (M n ) is a nontrivial 3group of exponent at most 9. Remark 1.4. Statement (i) is not explicitly stated in [Bo], but follows easily from the formula for the Betti numbers given in [Bo]. One can see the 3-torsion inH 1 (M 7 ) by looking at Figure 1.1. The union of the triangles shown is bounded by 3z where z = (13, 24) + (24, 15) + (15, 26) + (26, 13).Bouc shows that z is not a boundary; so z is a 3-torsion element.Friedman and Hanlon [FrHa] derive the following analogy of Theorem 1.3 (i), which settles the chessboard complex version of the conjecture in the case that n > 2m − 5, but leaves the conjecture unresolved the case that m ≤ n ≤ 2m − 5. Their result is a consequence of their formula for the Betti numbers of the chessboard complex derived in [FrHa] (see Theorem 6.1).Theorem 1.5 (Friedman and Hanlon [FrHa]). Let m ≤ n. Then the groupH νm,n (M m,n ) is finite if and only if n ≤ 2m − 5 and (m, n) / ∈ {(6, 6), (7, 7), (8, 9)}.In this paper we pick up where Bouc and Friedman-Hanlon left off. We prove the Björner-Lovász-Vrécica-Živaljević conjecture in the cases that were left unresolved in Bouc's work and Friedman-Hanlon's work (see Theorem 3.1). Moreover, we prove the following result which improves Theorem 1.3 by handling the remaining n ≡ 2 mod 3 case and making the exponent precise in all cases.Theorem 1.6. For n = 7, 10 or n ≥ 12 (except possibly n = 14), H νn (M n ) is a nontrivial elementary 3-group.We also prove the following analogous result for the chessboard complex.

We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented R \mathbb R -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.